SUBGROUPS OF SPIN(7) OR SO(7) WITH EACH ELEMENT CONJUGATE TO SOME ELEMENT OF G2 AND APPLICATIONS TO AUTOMORPHIC FORMS

As is well-known, the compact groups Spin(7) and SO(7) both have a single conjugacy class of compact subgroups of exceptional type G(2). We first show that if is a subgroup of Spin(7), and if each element of is conjugate to some element of G(2), then itself is conjugate to a subgroup of G(2). The analogous statement for SO(7) turns out be false, and our main result is a classification of all the exceptions. They are the following groups, embedded in each case in SO(7) in a very specific way: GL(2) (Z/3Z), SL2 (Z/3Z), Z/4Z x Z/2Z, as well as the nonabelian subgroups of GO(2) (C) with compact closure, similitude factors group {+/- 1}, and which are not isomorphic to the dihedral group of order 8. More generally, we consider the analogous problems in which the Euclidean space is replaced by a quadratic space of dimension 7 over an arbitrary field. This type of question naturally arises in some formulation of a converse statement of Langlands' global functoriality conjecture, to which the results above have thus some applications. Moreover, we give necessary and sufficient local conditions on a cuspidal algebraic regular automorphic representation of GL(7) over a totally real number field so that its associated l-adic Galois representations can be conjugate into G(2) ((Q(l)) over bar). We provide 11 examples over Q which are unramified at all primes.